In the wake of the recent and still ongoing deregulations of the electric power markets, load transmission and wheeling of power from distant generators to local load consumers has become common practice. As a consequence of the competition between utilities and the emerging need to optimize assets, substantially increased amounts of power are transmitted through the existing networks, invariably causing transmission bottlenecks and significant hourly variations of the generation and transmission pattern. This results in power transmission systems being operated ever closer to their stability boundaries and thus necessitates very accurate monitoring of the system's stability and real-time control mechanisms. Power systems in general can be viewed as non-linear hybrid systems, as they involve a combination of both continuous and discrete dynamics and corresponding control options.
Electric power transmission and distribution systems or networks comprise high-voltage tie lines for connecting geographically separated regions, medium-voltage lines, and substations for transforming voltages and for switching connections between lines. For managing the network, it is desirable to determine a state of the network, in particular load flows and stability margins. In recent times, not only root mean square (RMS) values of voltages, currents, active power and reactive power flowing in the network have been determined, but devices and systems for measuring voltage and current phasors at different locations of a network at exactly the same time have become available. The article “PMUs—A new approach to power network monitoring”, ABB Review 1/2001, p. 58, mentions a device called Phasor Measurement Unit (PMU) for accurate time-stamping of local power system information. A plurality of such phasor measurements collected from throughout the network at a central data processor in combination provide a snapshot of the overall electrical state of the power system.
The evolution in time of the overall state of the power system or a particular physical system quantity, such as the voltage at a certain node of a transmission network, is represented by a one—or multidimensional trajectory. Based on the current state of the system and taking into account potential control actions applied to the system, a future progression of the trajectory may be calculated. For instance, Model Predictive Control (MPC) is an academically and industrially well-known and accepted method for process control. The main principle can be seen from FIG. 1. A system model, representing e.g. a real power system and taking into account its dynamics, is used to predict output trajectories (xi) based on the current state at time t0 and for several different potential candidate input sequences (Δxi). A cost function is then defined based on the deviation of each predicted trajectory from a desired reference trajectory (xref) over a window in time called the prediction interval (tp). The optimal control, in the sense that it minimizes the defined cost function, is then obtained by solving an optimization problem.
There are two fundamentally different stages in MPC. Firstly there is a prediction stage which results in an approximation of the output trajectories for a certain input sequence. For linear systems this can be done by a number of matrix multiplications but for nonlinear systems this is usually done by simulation. Secondly there is a decision stage which typically consists of minimizing or maximizing a numerical performance objective which is based on the deviations of the trajectory approximation from a desired reference trajectory. Different methods have been applied such as linear/quadratic programming, nonlinear optimization or heuristic tree-search techniques. They all have in common that they require a large number of iterations, that is, evaluations of the cost criterion, which makes the computational burden of model-predictive control for large-scale nonlinear systems unattractive.
A technique based on trajectory sensitivities has been developed with the purpose of reducing the computational burden when the evaluation of multiple trajectories is necessary. Instead of evaluating all trajectories individually, only one trajectory is evaluated using a modified simulation method where the sensitivities with respect to key parameters are noted and approximations of the trajectories for such parameter changes can be made in a computationally efficient manner. The sensitivities of trajectories to initial conditions and/or parameters do provide an insight into the behaviour of a dynamic power system, as is described e.g. in the article by I. A. Hiskens and M. A. Pai, “Trajectory Sensitivity Analysis of Hybrid Systems”, IEEE Trans. Circuits and Systems, vol. 47, pp. 204-220, 2000. However, these capabilities of trajectory sensitivities have so far mainly been used for post-mortem analysis of a collapsed power system.